Question

Period of the function tanx + cotx is
$\begin{align}
  & a)\dfrac{\pi }{2} \\
 & b)\pi \\
 & c)\dfrac{3\pi }{2} \\
 & d)2\pi \\
\end{align}$

Answer 1

Hint: Now we know that the function tanx and cotx are both periodic functions and their period is $\pi $ . Now we know that if f(x) and g(x) are periodic functions with period \[{{T}_{1}}\] and ${{T}_{2}}$ respectively then f(x) + g(x) is periodic with period $LCM\left( {{T}_{1}},{{T}_{2}} \right)$ . Hence we can find the period of the given function.

Complete step-by-step answer:
Now let us first understand periodic functions.
A function f(x) is called a periodic function If it repeats the value after a regular interval.
Now formally we write it as a function f(x) is called a periodic function if f(x+T) = f(x). also in this case T is called the period of the function. Note that period is the distance between repetition.
Now for example we have that all the trigonometric functions are periodic.
Now sin, cos, sec, cosec are the functions with period $2\pi $ .
Tan and cot are the function with period $\pi $ .
Now note that this means.
Now addition of periodic function is also periodic function.
Hence if we have f(x) and g(x) is periodic then the function f(x) + g(x) is also periodic.
Also the period of f(x) + g(x) is given by LCM of period of f(x) and g(x).
Now since we have tanx and cotx are periodic with period $\pi $
We can say that tanx + cotx is also a periodic function and the period is LCM of $\left( \pi ,\pi \right)$ .
Hence we have the period of function tanx + cotx is $\pi $ .

So, the correct answer is “Option b”.

Note: Here note that when the functions are added the periods are not added but the period of the function is the resultant function is LCM of the two functions added. Also note that the period of all trigonometric ratios except tan and cot of $2\pi $ .